Multiple quadrature underintegrated finite elements

Wing Kam Liu*, Yu‐Kan ‐K Hu, Ted Belytschko

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

72 Scopus citations


New multiple‐quadrature‐point underintegrated finite elements with hourglass control are developed. The elements are selectively underintegrated to avoid volumetric and shear locking and save computational time. An approach for hourglass control is proposed such that the stabilization operators are obtained simply by taking the partial derivatives of the generalized strain rate vector with respect to the natural co‐ordinates so that the elements require no stabilization parameter. To improve accuracy over the traditional one‐point‐quadrature elements, several quadrature points are used to integrate the internal forces, especially for tracing the plastic fronts in the mesh during loading and unloading in elastic–plastic analysis. Two‐ and four‐point‐quadrature elements are proposed for use in the two‐ and three‐dimensional elements, respectively. Other multiple‐quadrature points can also be employed. Several numerical examples such as thin beam, plate and shell problems are presented to demonstrate the applicability of the proposed elements.

Original languageEnglish (US)
Pages (from-to)3263-3289
Number of pages27
JournalInternational Journal for Numerical Methods in Engineering
Issue number19
StatePublished - Oct 15 1994

ASJC Scopus subject areas

  • Numerical Analysis
  • General Engineering
  • Applied Mathematics


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